Chapter 3 Decision Analysis. Learning Objectives 1. List the steps of the decision-making process 2....

Chapter 3

Decision Analysis

Learning Objectives

1. List the steps of the decision-making process

2. Describe the types of decision-making environments

3. Make decisions under uncertainty4. Use probability values to make decisions

under risk

After completing this chapter, students will be able to:After completing this chapter, students will be able to:

Learning Objectives

1. Develop accurate and useful decision trees

2. Revise probabilities using Bayesian analysis

3. Use computers to solve basic decision-making problems

4. Understand the importance and use of utility theory in decision making

After completing this chapter, students will be able to:After completing this chapter, students will be able to:

Chapter Outline

3.1 Introduction3.2 The Six Steps in Decision Making3.3 Types of Decision-Making

Environments3.4 Decision Making under Uncertainty3.5 Decision Making under Risk3.6 Decision Trees3.7 How Probability Values Are

Estimated by Bayesian Analysis3.8 Utility Theory

Introduction

What is involved in making a good decision?

Decision theory is an analytic and systematic approach to the study of decision making

A good decision is one that is based on logic, considers all available data and possible alternatives, and the quantitative approach described here

The Six Steps in Decision Making

1. Clearly define the problem at hand2. List the possible alternatives3. Identify the possible outcomes or states

of nature4. List the payoff or profit of each

combination of alternatives and outcomes

5. Select one of the mathematical decision theory models

6. Apply the model and make your decision

© 2009 Prentice-Hall, Inc. 3 – 7

Other Methods

Non-quantitative methods Delphi Method

Delphi – ‘the Oracle’ in ancient Greece that gave predictions about the future

Finding a knowledgeable, wise person or persons to help make decisions

In-house methodsExecutive Opinion/DecisionMarketing Surveys

Thompson Lumber Company

Step 1 –Step 1 – Define the problem Expand by manufacturing and

marketing a new product, backyard storage sheds

Step 2 –Step 2 – List alternatives Construct a large new plant A small plant No plant at all

Step 3 –Step 3 – Identify possible outcomes The market could be favorable or

unfavorable


Step 4 –Step 4 – List the payoffs Identify conditional valuesconditional values for the

profits for large, small, and no plants for the two possible market conditions

Step 5 –Step 5 – Select the decision model Depends on the environment and

amount of risk and uncertaintyStep 6 –Step 6 – Apply the model to the data

Solution and analysis used to help the decision making


STATE OF NATURE

ALTERNATIVEFAVORABLE MARKET ($)

UNFAVORABLE MARKET ($)

Construct a large plant 200,000 –180,000

Construct a small plant 100,000 –20,000

Do nothing 0 0

Table 3.1

Types of Decision-Making Environments

Type 1:Type 1: Decision making under certainty Decision maker knows with certaintyknows with certainty the

consequences of every alternative or decision choice

Type 2:Type 2: Decision making under uncertainty The decision maker does not knowdoes not know the

probabilities of the various outcomesType 3:Type 3: Decision making under risk

The decision maker knows the knows the probabilitiesprobabilities of the various outcomes

Decision Making Under Uncertainty

1. Maximax (optimistic)

2. Maximin (pessimistic)

3. Criterion of realism (Hurwicz)

4. Equally likely (Laplace)

5. Minimax regret

There are several criteria for making decisions under uncertainty

Maximax

Used to find the alternative that maximizes the maximum payoff

Locate the maximum payoff for each alternative Select the alternative with the maximum

number

STATE OF NATURE



MAXIMUM IN A ROW ($)

Construct a large plant 200,000 –180,000 200,000

Construct a small plant 100,000 –20,000 100,000

Do nothing 0 0 0

Table 3.2

MaximaxMaximax

Maximin

Used to find the alternative that maximizes the minimum payoff

Locate the minimum payoff for each alternative Select the alternative with the maximum

number

STATE OF NATURE



MINIMUM IN A ROW ($)

Construct a large plant 200,000 –180,000 –180,000

Construct a small plant 100,000 –20,000 –20,000

Do nothing 0 0 0

Table 3.3 MaximinMaximin

Criterion of Realism (Hurwicz)

A weighted averageweighted average compromise between optimistic and pessimistic

Select a coefficient of realism Coefficient is between 0 and 1 A value of 1 is 100% optimistic Compute the weighted averages for each

alternative Select the alternative with the highest value

Weighted average = (maximum in row) + (1 – )(minimum in row)

Criterion of Realism (Hurwicz)

For the large plant alternative using = 0.8(0.8)(200,000) + (1 – 0.8)(–180,000) = 124,000

For the small plant alternative using = 0.8 (0.8)(100,000) + (1 – 0.8)(–20,000) = 76,000

STATE OF NATURE



CRITERION OF REALISM

( = 0.8)$



Do nothing 0 0 0

Table 3.4

RealismRealism

Equally Likely (Laplace)

Considers all the payoffs for each alternative Find the average payoff for each alternative Select the alternative with the highest average

STATE OF NATURE



ROW AVERAGE ($)



Do nothing 0 0 0Table 3.5

Equally likelyEqually likely

Minimax Regret

Based on opportunity lossopportunity loss or regretregret, the difference between the optimal profit and actual payoff for a decision

Create an opportunity loss table by determining the opportunity loss for not choosing the best alternative

Opportunity loss is calculated by subtracting each payoff in the column from the best payoff in the column

Find the maximum opportunity loss for each alternative and pick the alternative with the minimum number

Minimax Regret

STATE OF NATURE

FAVORABLE MARKET ($)


200,000 – 200,000 0 – (–180,000)

200,000 – 100,000 0 – (–20,000)

200,000 – 0 0 – 0

Table 3.6

Table 3.7

STATE OF NATURE



Construct a large plant 0 180,000

Construct a small plant 100,000 20,000

Do nothing 200,000 0

Opportunity Loss Tables

Minimax Regret

Table 3.8

STATE OF NATURE



MAXIMUM IN A ROW ($)

Construct a large plant 0 180,000 180,000

Construct a small plant 100,000 20,000 100,000

Do nothing 200,000 0 200,000MinimaxMinimax

© 2009 Prentice-Hall, Inc. 3 – 21

Who Are You?

Are you minimax regret? Are you maximax? A couple of points:

When people are in different situations they can make decisions differentlyE.g., At work vs. at home

As time goes on, a person may change in their attitudeE.g., Old people are more conservative

because they have no income and need to keep their money risk free.

Decision Making Under Risk

Decision making when there are several possible states of nature and we know the probabilities associated with each possible state

Most popular method is to choose the alternative with the highest expected monetary value (expected monetary value (EMVEMV))

EMV (alternative i) = (payoff of first state of nature)x (probability of first state of nature)+ (payoff of second state of nature)x (probability of second state of nature)+ … + (payoff of last state of nature)x (probability of last state of nature)

EMV for Thompson Lumber

Each market has a probability of 0.50 Which alternative would give the highest EMV? The calculations are

EMV (large plant) = (0.50)($200,000) + (0.50)(–$180,000)= $10,000

EMV (small plant) = (0.50)($100,000) + (0.50)(–$20,000)= $40,000

EMV (do nothing) = (0.50)($0) + (0.50)($0)= $0

EMV for Thompson Lumber

STATE OF NATURE


UNFAVORABLE MARKET ($) EMV ($)



Do nothing 0 0 0

Probabilities 0.50 0.50

Table 3.9 Largest Largest EMVEMV

Expected Value of Perfect Information (EVPI)

EVPI places an upper bound on what you should pay for additional information

EVPI = EVwPI – Maximum EMV EVwPI is the long run average return if we have

perfect information before a decision is made

EVwPI = (best payoff for first state of nature)x (probability of first state of nature)+ (best payoff for second state of nature)x (probability of second state of nature)+ … + (best payoff for last state of nature)x (probability of last state of nature)


Scientific Marketing, Inc. offers analysis that will provide certainty about market conditions (favorable)

Additional information will cost $65,000 Is it worth purchasing the information?


1. Best alternative for favorable state of nature is build a large plant with a payoff of $200,000Best alternative for unfavorable state of nature is to do nothing with a payoff of $0

EVwPI = ($200,000)(0.50) + ($0)(0.50) = $100,000

1. The maximum EMV without additional information is $40,000

EVPI = EVwPI – Maximum EMV= $100,000 - $40,000= $60,000


1. Best alternative for favorable state of nature is build a large plant with a payoff of $200,000Best alternative for unfavorable state of nature is to do nothing with a payoff of $0

EVwPI = ($200,000)(0.50) + ($0)(0.50) = $100,000

1. The maximum EMV without additional information is $40,000

EVPI = EVwPI – Maximum EMV= $100,000 - $40,000= $60,000

So the maximum Thompson should pay for the additional information is $60,000

Expected Opportunity Loss

Expected opportunity lossExpected opportunity loss (EOL) is the cost of not picking the best solution

First construct an opportunity loss table For each alternative, multiply the

opportunity loss by the probability of that loss for each possible outcome and add these together

Minimum EOL will always result in the same decision as maximum EMV

Minimum EOL will always equal EVPI

Expected Opportunity Loss

EOL (large plant) = (0.50)($0) + (0.50)($180,000)= $90,000

EOL (small plant) = (0.50)($100,000) + (0.50)($20,000)= $60,000

EOL (do nothing) = (0.50)($200,000) + (0.50)($0)= $100,000

Table 3.10

STATE OF NATURE


UNFAVORABLE MARKET ($) EOL

Construct a large plant 0 180,000 90,000

Construct a small plant 100,000 20,000 60,000

Do nothing 200,000 0 100,000

Probabilities 0.50 0.50 Minimum Minimum EOLEOL

Sensitivity Analysis

Sensitivity analysis examines how our decision might change with different input data

For the Thompson Lumber example

P = probability of a favorable market

(1 – P) = probability of an unfavorable market


EMV(Large Plant) = $200,000P – $180,000)(1 – P)= $200,000P – $180,000 + $180,000P= $380,000P – $180,000

EMV(Small Plant) = $100,000P – $20,000)(1 – P)= $100,000P – $20,000 + $20,000P= $120,000P – $20,000

EMV(Do Nothing) = $0P + 0(1 – P)= $0


$300,000

$200,000

$100,000

0

–$100,000

–$200,000

EMV Values

EMV (large plant)

EMV (small plant)

EMV (do nothing)

Point 1

Point 2

.167 .615 1

Values of P

Figure 3.1


Point 1:Point 1:EMV(do nothing) = EMV(small plant)

000200001200 ,$,$ P 167000012000020

.,,

P

00018000038000020000120 ,$,$,$,$ PP

6150000260000160

.,,

P

Point 2:Point 2:EMV(small plant) = EMV(large plant)


$300,000

$200,000

$100,000

0

–$100,000

–$200,000

EMV Values

EMV (large plant)

EMV (small plant)

EMV (do nothing)

Point 1

Point 2

.167 .615 1

Values of P

Figure 3.1

BEST ALTERNATIVE

RANGE OF P VALUES

Do nothing Less than 0.167

Construct a small plant 0.167 – 0.615

Construct a large plant Greater than 0.615

Using Excel QM to Solve Decision Theory Problems

Program 3.1A

Using Excel QM to Solve Decision Theory Problems

Program 3.1B

Decision Trees Any problem that can be presented in a

decision table can also be graphically represented in a decision treedecision tree

Decision trees are most beneficial when a sequence of decisions must be made

All decision trees contain decision pointsdecision points or nodesnodes and state-of-nature pointsstate-of-nature points or nodesnodes A decision node from which one of several

alternatives may be chosen A state-of-nature node out of which one state

of nature will occur

Five Steps toDecision Tree Analysis

1. Define the problem2. Structure or draw the decision tree3. Assign probabilities to the states of

nature4. Estimate payoffs for each possible

combination of alternatives and states of nature

5. Solve the problem by computing expected monetary values (EMVs) for each state of nature node

Structure of Decision Trees

Trees start from left to right Represent decisions and outcomes in

sequential order Squares represent decision nodes Circles represent states of nature nodes Lines or branches connect the decisions

nodes and the states of nature

Thompson’s Decision Tree

Favorable Market

Unfavorable Market

Favorable Market

Unfavorable Market

Do Nothing

Construct

Large P

lant

1

Construct

Small Plant2

Figure 3.2

A Decision Node

A State-of-Nature Node

Thompson’s Decision Tree

Favorable Market

Unfavorable Market

Favorable Market

Unfavorable Market

Do Nothing

Construct

Large P

lant

1

Construct

Small Plant2

Alternative with best EMV is selected

Figure 3.3

EMV for Node 1 = $10,000

= (0.5)($200,000) + (0.5)(–$180,000)

EMV for Node 2 = $40,000

= (0.5)($100,000) + (0.5)(–$20,000)

Payoffs

$200,000

–$180,000

$100,000

–$20,000

$0

(0.5)

(0.5)

(0.5)

(0.5)

Thompson’s Complex Decision Tree

First Decision Point

Second Decision Point

Favorable Market (0.78)

Unfavorable Market (0.22)











Large Plant

Small Plant

No Plant

6

7

Con

duct

Mar

ket S

urve

y

Do Not Conduct Survey

Large Plant

Small Plant

No Plant

2

3

Large Plant

Small Plant

No Plant

4

5

1Results

Favorable

ResultsNegative

Survey (0

.45)

Survey (0.55)

Payoffs

–$190,000

$190,000

$90,000

–$30,000

–$10,000

–$180,000

$200,000

$100,000

–$20,000

$0

–$190,000

$190,000

$90,000

–$30,000

–$10,000

Figure 3.4


1.1. Given favorable survey results,EMV(node 2) = EMV(large plant | positive survey)

= (0.78)($190,000) + (0.22)(–$190,000) = $106,400EMV(node 3) = EMV(small plant | positive survey)

= (0.78)($90,000) + (0.22)(–$30,000) = $63,600EMV for no plant = –$10,000

2.2. Given negative survey results,EMV(node 4) = EMV(large plant | negative survey)

= (0.27)($190,000) + (0.73)(–$190,000) = –$87,400EMV(node 5) = EMV(small plant | negative survey)

= (0.27)($90,000) + (0.73)(–$30,000) = $2,400EMV for no plant = –$10,000


3.3. Compute the expected value of the market survey,EMV(node 1) = EMV(conduct survey)

= (0.45)($106,400) + (0.55)($2,400)= $47,880 + $1,320 = $49,200

4.4. If the market survey is not conducted,EMV(node 6) = EMV(large plant)

= (0.50)($200,000) + (0.50)(–$180,000) = $10,000EMV(node 7) = EMV(small plant)

= (0.50)($100,000) + (0.50)(–$20,000) = $40,000EMV for no plant = $0

5.5. Best choice is to seek marketing information


Figure 3.4

First Decision Point

Second Decision Point













Large Plant

Small Plant

No Plant

Con

duct

Mar

ket S

urve

y

Do Not Conduct Survey

Large Plant

Small Plant

No Plant

Large Plant

Small Plant

No Plant

Results

Favorable

ResultsNegative

Survey (0

.45)

Survey (0.55)

Payoffs

–$190,000

$190,000

$90,000

–$30,000

–$10,000

–$180,000

$200,000

$100,000

–$20,000

$0

–$190,000

$190,000

$90,000

–$30,000

–$10,000

$40,000$2,400

$106,400

$49,200

$106,400

$63,600

–$87,400

$2,400

$10,000

$40,000

Expected Value of Sample Information

Thompson wants to know the actual value of doing the survey

EVSI = –

Expected valuewithwith sample

information, assumingno cost to gather it

Expected valueof best decisionwithoutwithout sample

information

= (EV with sample information + cost)– (EV without sample information)

EVSI = ($49,200 + $10,000) – $40,000 = $19,200


How sensitive are the decisions to changes in the probabilities? How sensitive is our decision to the

probability of a favorable survey result? That is, if the probability of a favorable

result (p = .45) where to change, would we make the same decision?

How much could it change before we would make a different decision?


p = probability of a favorable survey result(1 – p) = probability of a negative survey resultEMV(node 1) = ($106,400)p +($2,400)(1 – p)

= $104,000p + $2,400

We are indifferent when the EMV of node 1 is the same as the EMV of not conducting the survey, $40,000

$104,000p + $2,400 = $40,000$104,000p = $37,600p = $37,600/$104,000 = 0.36

Bayesian Analysis

Many ways of getting probability data It can be based on

Management’s experience and intuition Historical data Computed from other data using Bayes’

theorem Bayes’ theorem incorporates initial

estimates and information about the accuracy of the sources

Allows the revision of initial estimates based on new information

Calculating Revised Probabilities

In the Thompson Lumber case we used these four conditional probabilities

P (favorable market(FM) | survey results positive) = 0.78P (unfavorable market(UM) | survey results positive) = 0.22

P (favorable market(FM) | survey results negative) = 0.27P (unfavorable market(UM) | survey results negative) = 0.73

The prior probabilities of these markets are

P (FM) = 0.50P (UM) = 0.50


Through discussions with experts Thompson has learned the following

He can use this information and Bayes’ theorem to calculate posterior probabilities

STATE OF NATURE

RESULT OF SURVEY

FAVORABLE MARKET (FM)

UNFAVORABLE MARKET (UM)

Positive (predicts favorable market for product)

P (survey positive | FM) = 0.70

P (survey positive | UM) = 0.20

Negative (predicts unfavorable market for product)

P (survey negative | FM) = 0.30

P (survey negative | UM) = 0.80

Table 3.11


Recall Bayes’ theorem is

)()|()()|()()|(

)|(APABPAPABP

APABPBAP

whereevents two anyBA,

AA of complement

For this example, A will represent a favorable market and B will represent a positive survey


P (FM | survey positive)

P(UM)|UM)P(P(FM) |FM)P(FMPFMP

positive surveypositive survey

positive survey )()|(

780450350

500200500700500700

...

).)(.().)(.().)(.(

P(FM)|FM)P(P(UM) |UM)P(UMPUMP

positive surveypositive survey

positive survey )()|(

220450100

500700500200500200

...

).)(.().)(.().)(.(

P (UM | survey positive)


POSTERIOR PROBABILITY

STATE OF NATURE

CONDITIONAL PROBABILITY

P(SURVEY POSITIVE | STATE

OF NATURE)PRIOR

PROBABILITYJOINT

PROBABILITY

P(STATE OF NATURE | SURVEY

POSITIVE)

FM 0.70 X 0.50 = 0.35 0.35/0.45 = 0.78

UM 0.20 X 0.50 = 0.10 0.10/0.45 = 0.22

P(survey results positive) = 0.45 1.00

Table 3.12


P (FM | survey negative)

P(UM)|UM)P(P(FM) |FM)P(FMPFMP

negative surveynegative survey

negative survey )()|(

270550150

500800500300500300

...

).)(.().)(.().)(.(

P(FM)|FM)P(P(UM) |UM)P(UMPUMP

negative surveynegative survey

negative survey )()|(

730550400

500300500800500800

...

).)(.().)(.().)(.(

P (UM | survey negative)


POSTERIOR PROBABILITY

STATE OF NATURE

CONDITIONAL PROBABILITY

P(SURVEY NEGATIVE | STATE

OF NATURE)PRIOR

PROBABILITYJOINT

PROBABILITY

P(STATE OF NATURE | SURVEY

NEGATIVE)

FM 0.30 X 0.50 = 0.15 0.15/0.55 = 0.27

UM 0.80 X 0.50 = 0.40 0.40/0.55 = 0.73

P(survey results positive) = 0.55 1.00

Table 3.13

Potential Problems Using Survey Results

We can not always get the necessary data for analysis

Survey results may be based on cases where an action was taken

Conditional probability information may not be as accurate as we would like

Utility Theory

Monetary value is not always a true indicator of the overall value of the result of a decision

The overall value of a decision is called utilityutility

Rational people make decisions to maximize their utility

Heads (0.5)

Tails (0.5)

$5,000,000

$0

Utility Theory

Accept Offer

Reject Offer

$2,000,000

EMV = $2,500,000

Figure 3.6

Utility Theory

Utility assessmentUtility assessment assigns the worst outcome a utility of 0, and the best outcome, a utility of 1

A standard gamblestandard gamble is used to determine utility values

When you are indifferent, the utility values are equal

Expected utility of alternative 2 =Expected utility of alternative 1Utility of other outcome = (p)(utility of best outcome, which is 1)

+ (1 – p)(utility of the worst outcome, which is 0)Utility of other outcome = (p)(1) + (1 – p)(0) = p

Standard Gamble

Best OutcomeUtility = 1

Worst OutcomeUtility = 0

Other OutcomeUtility = ?

(p)

(1 – p)

Alternativ

e 1

Alternative 2

Figure 3.7

Investment Example Jane Dickson wants to construct a utility curve

revealing her preference for money between $0 and $10,000

A utility curve plots the utility value versus the monetary value

An investment in a bank will result in $5,000 An investment in real estate will result in $0 or

$10,000 Unless there is an 80% chance of getting $10,000

from the real estate deal, Jane would prefer to have her money in the bank

So if p = 0.80, Jane is indifferent between the bank or the real estate investment

Investment Example

Figure 3.8

p = 0.80

(1 – p) = 0.20

Invest in

Real Estate

Invest in Bank

$10,000U($10,000) = 1.0

$0U($0.00) = 0.0

$5,000U($5,000) = p = 1.0

Utility for $5,000 = U($5,000) = pU($10,000) + (1 – p)U($0)= (0.8)(1) + (0.2)(0) = 0.8

Investment Example

Utility for $7,000 = 0.90Utility for $3,000 = 0.50

We can assess other utility values in the same way For Jane these are

Using the three utilities for different dollar amounts, she can construct a utility curve

Utility Curve

U ($7,000) = 0.90

U ($5,000) = 0.80

U ($3,000) = 0.50

U ($0) = 0

Figure 3.9

1.0 –

0.9 –

0.8 –

0.7 –

0.6 –

0.5 –

0.4 –

0.3 –

0.2 –

0.1 –

| | | | | | | | | | |

$0 $1,000 $3,000 $5,000 $7,000 $10,000

Monetary Value

Utilit

y

U ($10,000) = 1.0

Utility Curve

Jane’s utility curve is typical of a risk avoider A risk avoider gets less utility from greater risk Avoids situations where high losses might

occur As monetary value increases, the utility curve

increases at a slower rate A risk seeker gets more utility from greater risk As monetary value increases, the utility curve

increases at a faster rate Someone who is indifferent will have a linear

utility curve

Utility Curve

Figure 3.10Monetary Outcome

Utilit

y

Risk Avoider

Risk

Indiff

eren

ce

Risk Seeker

Utility as a Decision-Making Criteria

Once a utility curve has been developed it can be used in making decisions

Replace monetary outcomes with utility values

The expected utility is computed instead of the EMV


Mark Simkin loves to gamble He plays a game tossing thumbtacks in

the air If the thumbtack lands point up, Mark wins

$10,000 If the thumbtack lands point down, Mark

loses $10,000 Should Mark play the game (alternative 1)?


Figure 3.11

Tack Lands Point Up (0.45)

Alternativ

e 1

Mark Plays the Game

Alternative 2

$10,000

–$10,000

$0

Tack Lands Point Down (0.55)

Mark Does Not Play the Game


Step 1– Define Mark’s utilities

U (–$10,000) = 0.05U ($0) = 0.15

U ($10,000) = 0.30

Step 2 – Replace monetary values with

utility valuesE(alternative 1: play the game) = (0.45)(0.30) + (0.55)(0.05)

= 0.135 + 0.027 = 0.162E(alternative 2: don’t play the game)= 0.15


Figure 3.12

1.00 –

0.75 –

0.50 –

0.30 –0.25 –

0.15 –

0.05 –0 –| | | | |

–$20,000 –$10,000 $0 $10,000 $20,000Monetary Outcome

Utilit

y


Figure 3.13

Tack Lands Point Up (0.45)

Alternativ

e 1

Mark Plays the Game

Alternative 2

0.30

0.05

0.15

Tack Lands Point Down (0.55)

Don’t Play

UtilityE = 0.162

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